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Manipulation of ultracold Bose gases in a time-averaged orbiting potential
Cleary, P.W.
Publication date
2012
Link to publication
Citation for published version (APA):
Cleary, P. W. (2012). Manipulation of ultracold Bose gases in a time-averaged orbiting
potential.
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Chapter 2
Experimental Set-up
2.1
Introduction
The essential elements of a BEC experiment are a trapped cloud of atoms under vacuum conditions, laser light for optical cooling and trapping, an adjustable form of magnetic trapping, a split-second control system and a detection method of the cloud of atoms. These aspects were in place from a set of three predecessors who developed the existing apparatus [38, 39, 44] and much of this apparatus could be used unchanged or with minor modifications, namely the vacuum system and magnetic trapping apparatus. The optical system required substantial improvement and these improvements are discussed in Chapter 3.
In reaching condensation, a good vacuum is required to ensure that at least some of the atoms which are being cooled to condensation are not heated by collisions with background "thermal" atoms before reaching BEC. In this set-up, the conundrum of desiring excellent vacuum conditions and yet requiring an adequate vapour pressure as a source of cold atoms was solved by using a dual vacuum chamber system [38]. Optical cooling and trapping is performed with resonant frequency-stabilized laser light to form a two-dimensional magneto-optical trap (2D MOT) [45] in the lower vacuum chamber (using bulk optics) and a three-dimensional magneto-optical trap (3D MOT) in the upper (ultra high vacuum) chamber (using an optical fiber distrib-utor). Laser light is then utilized to pump the atoms to states suitable for magnetic trapping. The magnetic trapping is performed in an adjustable Ioffe-Pritchard (IP) trap suitable to cool the atoms by rf evaporation to BEC. The ultra-cold atomic cloud is detected by time-of-flight absorption imaging on to a CCD camera after release from the trap. The experiments are controlled with the aid of a LabView graphics user interface connected to National Instrument input and output cards. The software was running on a regular Windows XP PC.
In this chapter the various components are discussed, with emphasis on changes to the set-up with respect to the previous thesis works on this apparatus. After a description of the vacuum system in Section 2.2 we discuss the various optical
frequencies used in the experiment along with the organization of the optical table (Section 2.3). In Section 2.4 we discuss the 2D MOT system with emphasis on the advantage of the use of permanent magnets for the creation of magnetic field gradients. A complete overhaul of the 3D MOT is presented in Section 2.5. The Ioffe-Pritchard trap and its current supply are introduced in Section 2.6. This is followed by a discussion of the implementation of time-averaged orbiting potentials (Section 2.7) as well as rf evaporative cooling (Section 2.8). The chapter is concluded with an overview of the components used for interfacing in Section 2.9.
2.2
Vacuum System
Vacuum conditions are essential in order to reach Bose-Einstein condensation with a relatively large number of atoms and for this condensate to a workable lifetime. The process of enhancing the phase space density to condensation by evaporative cooling takes about 10 s and another few seconds are needed for the experiments. The required lifetime was realized by separating the source of cold atoms from the measurement region by the use of a differentially pumped dual vacuum chamber system. As can be seen in Fig. 2.1, one vacuum chamber is located directly above the other and there is a channel of diameter 0.8 mm and length 3 mm joining the two chambers. Differential pumping is performed via this channel and allows us to achieve a vacuum of 10−10 mbarin the upper chamber and yet have a Rb vapour
pressure of 10−7mbarin the lower chamber. The collisional lifetime of trapped atoms
due to collisions with background gas in the upper chamber is 50 s. Each of these two quartz chambers is joined to a metal manifold by a pair of viton rubber O-rings and the space between these is pumped by a rotary pump, which greatly reduces permeation through the O-rings (by 6 orders of magnitude [38]). The entire vacuum set-up rests on a breadboard at a height of some 500 mm above of the optical table. The vapour pressure of rubidium in the lower chamber was created when the first BECs were made in this machine. It was supplied by heating the rubidium reservoir after which the supply valve was closed. Since then rubidium has been in overabundance in this chamber so that there is always a Rb vapour pressure of ∼ 5 × 10−7mbar,the saturated vapour pressure of rubidium at room temperature [46]. The
leak rate to the upper chamber via the adjoining channel is calculated to be negligible compared to the controlled flux which will be supplied during measurements.
At the beginning of this work, the entire apparatus was dismantled and trans-ported from the FOM institute AMOLF to the Van der Waals-Zeeman Institute (WZI) of the University of Amsterdam (UvA). During this transport and subsequent storage, for some days the system was not pumped and the quality of vacuum in the UHV (upper) vacuum chamber deteriorated several orders of magnitude. Placing a backing turbo pump (Balzers) behind the ion pump, over a period of 48 hours, the vacuum recovered from 9 × 10−6mbarto 5 × 10−9mbaras measured on an ion gauge.
Figure 2.1: Schematic of the vacuum system showing upper and lower vapour cells, the differential pumping hole and all connections to pumps, gauges and the rubidium reservoir.
Both ion pumps were subsequently baked out and over a period of one month vacuum pressure in the upper chamber recovered to 10−11 mbar.
While the upper vacuum system was pumped directly by an 40 l/s ion pump (Varian) and was connected to an ion gauge (Varian), the lower cell was pumped via the differential pumping hole and the O-rings between the chambers pumped by a rotary pump. In the lower cell the rubidium pressure remained at the saturated vapour pressure. However, contamination presumably by air permeating the viton O-rings (during the move when the rotary pump was without power) resulted in an increasing amount of a greyish compound attributed to rubidium hydroxide. This contamination adhered to the windows of optical access of the lower cell reducing transmission at the lower half of the cell, but not affecting the quality of the vacuum. Various attempts were made to remediate this problem. The system was baked out repeatedly up to 135 C. Beyond this temperature the viton O-rings from the cells to the metal manifold tend to malfunction due to thermal deformation and unfortunately this temperature was not high enough to desorb the hydroxide from the glass. Desorption techniques for Rubidium on glass using white light [47] and blue LEDs [48] also failed to remove the hydroxide. In order to avoid a repeat of this contamination situation a valve was placed between the O-rings and the rotary pump.
2.3
Optics
In this section a summary is given of the optical frequencies employed for running the experiment. The relevant optical transitions are indicated in Fig. 2.2. The electronic ground state 52S
1/2of87Rb has two hyperfine levels 52S1/2(F = 1)and 52S1/2(F = 2)
separated by 6.8 GHz. Optical transitions are possible between the ground state 52S
1/2 and the first two excited states 52P1/2 and 52P3/2 coupling these levels with
what are known as the D1 and D2 lines, respectively. Only D2 transitions are used and indicated in Fig. 2.2. Electric-dipole transitions are allowed between the levels 52S
1/2 and 52P3/2 for ΔF = F − F = −1, 0, +1. The excited state 52P3/2 has four
hyperfine levels F = 0, 1, 2, 3 so for the D2 line a total of six hyperfine transitions are possible, three from F = 1 and three from F = 2. In our experiments we use two of these from F = 1 and two from F = 2. For locking the frequency of the master laser we use the crossover transition F = 2 → F = 1, 3. The lasers are used for the following purposes:
Laser cooling: Laser cooling makes use of near resonant light to reduce the mo-mentum of atoms by the absorption of photons. Each photon carries momo-mentum E = hωand so can effect a change of velocity of hk/m on the atom which absorbs its energy. By arranging light beams such that the momentum of an ensemble of atoms is reduced in every direction, the average kinetic energy can be reduced and so the ensemble cooled. This can be accomplished with the aid of the Doppler shift, which
D2 780 nm 52P 3/2 52S 1/2 F=1 F=2 F’=0 F’=1 F’=2 F’=3 6.8 GHz 72 MHz 157 MHz 212 Mhz co F=1,3 Maste r lase r (crosso ver)
repumping MOT cooling
F=2 op tical pumping F=1 op tical pumping 267 Mhz 12 MHz
Figure 2.2: Energy level diagram for87Rb showing the optical transitions and colour coding used in this thesis. Note that the frequency splittings are not drawn to scale.
makes the light force velocity dependent. This type of laser cooling is called Doppler cooling and has a limit known as the Doppler limit. Laser cooling is used here in an optical molasses configuration with laser cooling beams from all 6 directions and in a magneto-optical trap as described in Sections 2.4 and 2.5. A review of laser cooling is included in [11]. To produce a considerable cooling each atom must be excited many times. Therefore, the cooling is performed with light red-detuned from the σ+
cycling transition |F = 2, mF = 2 → |F = 3, mF = 3 of the D2 line (see Fig. 2.2).
Repumping: Because of the relatively small frequency differences of the levels in the 52P
3/2 manifold (see Fig. 2.2), atoms in the F = 2 state have an off-resonant
probability of being spuriously excited by the cooling light to the F = 2 level from which they can decay to the F = 1 ground state hyperfine level. As excitation from the F = 1 state is negligible at the cooling frequency, the F = 1 state is a dark state in which the atoms are trapped. The trapping is avoided by the addition of a second laser (repumper) on the F = 1 → F = 2 transition in order to enable decay back to the F = 2 ground state.
Optical Pumping: Resonant light is also used to spin polarize the sample. This is done by optically pumping to one of the desired magnetically trappable states, |F, mF = |2, 2 and |F, mF = |1, −1 . Pumping to the |2, 2 state is done with
30 dB P.D. Cavity λ/2 30 dB 30 dB 60 dB λ/2 AO M λ/4 AOM 30 dB 30 dB AO M AO M λ/4 AO M upper MOT (via octopus) Opt pumping F=1 Imaging F=2 Imaging F=1 Opt pumping F=2 Ȟ2IN Push beam Lower MOT optics TA 3 TA 2 Slave laser Ȟ1IN Ȟ3IN
Figure 2.3: Schematic diagram showing positioning of optical elements and light paths and frequencies of laser beams on main table. Incoming fibers 1, 2, 3 carry light stabilized at, respectively, the crossover transition (ν1, shown in blue), the repumper frequency (ν2, red)
and the MOT cooling frequency (ν3, green). Light leaves the table to the lower MOT,
upper MOT (via octopus), and to both the imaging and optical pumping fibers for F = 1 and F = 2, respectively.
dark). Similarly, optical pumping to the |1, −1 level is done with σ− light on the
F = 1→ F = 1 transition (for which the |1, −1 state is dark). Preparation of the |1, −1 state is discussed in detail in Section 4.2.1.
Detection: The sample is detected using time-of-flight absorption imaging onto a CCD camera after release from the magnetic trap using a closed-cycle transition. The choice of imaging transition for F = 1 and F = 2 atoms is discussed in Section 4.2.1.
Optical beam delivery For the optical system it was decided to have a complete redesign. The two stabilized master lasers (at frequencies ν1 and ν2) were removed
from the main table and installed on side tables as part of the modularization of the experiment (see Chapter 3). This greatly improved the stability and reliability of the optical system and strongly reduced the number of optical elements on the
main table. Key to the process of modularization was the use of 4 polarization maintaining fibers to deliver light from laser modules on one of the well-isolated side tables to the main table optical system. The arrangement of the main table is shown in Fig. 2.3. Fiber 1, 2, 3 carry light stabilized at, respectively, the crossover transition (ν1, shown in blue), the repumper frequency (ν2, red) and the MOT cooling frequency
(ν3= ν1+ 200 MHz, green).
Fiber 1 is used to inject a slave laser similar to that described in [49] monitored by Fabry-Perot cavity. This light at frequency ν1is then split at a polarization cube.
One part is sent through a fixed-frequency AOM and coupled into a fiber for optical pumping on the F = 2 → F = 2 transition at frequency ν = ν1− 55 MHz (dark blue
path in Fig. 2.3). The other is sent double pass through an AOM and into a fiber for imaging atoms at the |2, 2 −→ |3, 3 transition. By adjusting the control voltage of this AOM the detuning of the imaging light can be set without losing alignment. Thus we can image F = 1 at detunings from −60 MHz to +20 MHz relative to the imaging transition at ν = ν1+ 212 MHz.
Fiber 2 is used to inject an in-house designed tapered amplifier at frequency ν2
(TA 2 - see Section 3.1). The amplified light is spatially filtered by a fiber and then split four ways to supply power for (a) repumping the upper MOT (via the Octopus fiber distribution system - see Section 2.5); (b) repumping the lower MOT (via solid optics); (c) optical pumping to the |1, −1 state using a fixed AOM (ν = ν2−157 MHz,
orange path in Fig. 2.3); (d) imaging F = 1 atoms using a variable double pass AOM. In the latter case we scan between −20 MHz and +40 MHz relative to the transition |F = 1, mF = 1 −→ |F = 2, mF = 2 at frequency ν2.
Fiber 3 is used to inject a tapered amplifier at frequency ν3(TA 3) and combined
with repumper light at frequency ν2 to form the four main beams and single push
beam of the lower MOT. A shutter in front of each incoming fiber ensures absolute extinction when the light is not use, greatly reducing stray light and thus increasing the trapping lifetimes of the samples. The absence of frequency stabilized lasers on the main optical table means that acoustic noise caused by these shutters is not a problem. The components on the main optical table are much less sensitive to such noise.
2.4
2D MOT
The87Rb atoms are optically cooled from room temperature to sub-millikelvin
tem-peratures in a two-dimensional magneto-optical trap (2D MOT) acting as a source of cold atoms. This 2D MOT (the lower MOT) consists of two orthogonal pairs of hor-izontal light beams and a 2D quadrupole field with its symmetry axis in the vertical direction. Each beam pair consists of two cooling beams of opposite helicity and light for repumping. As the light beams trap and cool the atoms horizontally this results in an vertically elongated MOT cloud. In the vertical direction a beam from below
Permanent magnets
Ioffe coils
Platform for MOT alignment
85 mm
push beam
Figure 2.4: Left: old 2D MOT with Ioffe coils. Right: the new set-up with permanent magnets (see Fig 2.5) in the configuration shown in Fig 2.6.
pushes the atoms through the differential pumping channel towards the 3D MOT in the upper vacuum chamber. This channel also acts as a velocity selector for the atoms of the 2D MOT; only atoms with a sufficiently low transverse velocity pass through [45]. The magnetic trap previously used for the 2D MOT consisted of four race-track shaped coils generating the 2D quadrupole field with its symmetry axis vertically along the central axis of the cell. This system pre-existed from previous projects on this machine and was combined with retroreflected cooling beams. This set-up was shown to be a successful rival to alternative sources of precooled atoms such as a Zeeman slower [45]. In view of the contamination of the lower vacuum system men-tioned in Section 2.2 it proved to be imperative to use four balanced MOT beams to compensate for the reduction in cell transparency to 70−75%. The increased amount of stray light made locating the MOT from a side-on view impossible and this greatly complicated the alignment procedure of the MOT coils. A convenient solution was found by using an alternative and much simpler method of creating a quadrupole field by the use of permanent magnets, see Fig. 2.4. Proof that a 2D quadrupole field can be formed which just two dipoles or bar magnets is given in Appendix A. To produce a 2D quadrupole field which is homogenous over a distance of 5 cm in the
40mm
z x
z y
Figure 2.5: Left: diagram of magnet holder showing dimensions and positioning of perma-nent magnets 1-6. Right: magnet holders are placed opposite to one another to create a quadropole field with a linear field minimum (see Fig. 2.6).
z direction, we placed a set of three bar magnets on a holder as shown in Fig 2.6. Summing the contributions of all six magnets n = 1, · · · , 6 in the arrangement shown in Fig. 2.5 we obtain the total magnetic field at position r,
Bi(r) = 6
n=1
(−1)nBi(r, rn) with i ∈ {x, y, z}. (2.1)
This equation can used along with the results in Appendix A to calculate the magnetic field close to the axis of the cell.
Measurements of magnets and implementation
The magnetization of the bar magnets was found with the aid of Eq. (A.7) by taking an individual magnet bar of 25 × 10 × 3 mm (Eclipse Magnetics Ltd.) and measuring the magnetic field as a function of the distance from the magnet center. The field was measured with a Hall probe (Honeywell SS490 ) calibrated against a precision solenoid carrying a known current. Two individual magnets measured in this way showed good agreement with one another. The value for the magnetization found is reported in [50] as 879 kA/m and lies well within the range of values 850 − 950 kA/m reported in the literature. This value was then used to calculate the field along the z-axis between two assemblies of 3 magnets each. Comparing this calculation to the measurements of the field from each of the three-bar magnet assemblies is shown in Fig. 2.7.
Light beam Light beam x axis y axis Magnet 1,3,5 Magnet 2,4,6
Figure 2.6: Schematic illustration of the 2D MOT arrangement as seen along the vertical symmetry axis of the lower vacuum cell. The solid square shows the orientation of the cell. The magnet sets (1, 3, 5) and (2, 4, 6) are placed on one of the cell diagonals on either side of the symmetry axis. The curved lines shows the direction of the 2D magnetic quadrupole field lines close to the axis.
It should be noted that the magnets were tested well beyond the limits of what was required for the experiments. Using the magnet assemblies it was possible to create a quadrupole magnetic field which was constant in the axial direction to within 3% over a distance of 5 cm - similar to the length over which the MOT beams are applied in the vertical direction. The magnet holders were placed on translation stages along the diagonal of the cell and adjusted to 42.5 mm distance from the centre of the cell (see Fig. 2.4) to give a gradient of 23 G/cm. Four beams with 22 mW of cooling light each were combined with 1 mW repumper power and transmitted through the cell using the cylindrical optics to produce elliptical beams as described by Dieckmann [45]. The opposing beams were aligned on top of each other. Precision alignment of the atomic beam was carried out by placing a camera below the cell. This camera was aligned to the axis of the differential pumping channel by shining white light from above. With a clean line of sight in place, the 2D MOT was then aligned to the bright spot from the top using the translation stages of the permanent magnet holders. Using this method a beam of atoms to the upper MOT could be established albeit with a low flux (107 atoms per second) as measured by capture in the upper (3D) MOT.
This flux could be tweaked with the translation stages but a substantial increase in flux was found only after the installation of a push beam from the bottom of the cell, also taken from the trapping light of TA 3 (see Fig. 2.3). This beam was reflected via a mirror at 45◦ to the vertical located directly below the cell (Fig. 2.4) but allowing
-15 -10 -5 0 5 10 15 0 2 4 6 8 10 12 14 Magnet holder 1 Magnet holder 2 Calculated Ma gn e ti c fi e ld [G au ss ] Position on z-axis [cm] -15 -10 -5 0 5 10 15 0 2 4 6 8 10 12 14
Magnetic Field (Gauss)
Position on z-axis (cm)
Figure 2.7: Measurements of magnetic field from each of the magnet sets (upper and lower points) at a distance of 4 cm from the magnet edge (see Fig. 2.5), along the long axis of the cell (z-axis) showing homogenenity over a distance of 5 cm.The mismatch in the wings of the distribution may be due to small measurement errors with the Hall probe.
The push-beam increased the flux to 5 × 108 atoms per second and the loading time
of the 3D MOT reduced to about 5 s comfortably increasing the repetition rate of experiments. The flux was found to be maximal at a detuning of 12 MHz to the red of the cooling transition in zero field. The quadrupole field drops rapidly with distance and the permanent magnets had no noticeable effect on the performance of the 3D MOT or the experiments described in this thesis. Permanent magnets were also successfully used in our laboratory for 2D MOT of other species such as K and Li [51].
2.5
3D MOT
The cooled atoms from the 2D MOT which pass through the differential pumping channel are then captured in a 3D MOT. This MOT consists of a set of six beams which overlap at the zero of a 3D magnetic quadrupole field. The quadrupole field is provided by the pinch coils of the Ioffe Pritchard trap (to be described in Section 2.6). This combination produces a spherically shaped cloud which is trapped and cooled in all directions and appropriate for transfer into a magnetic trap such as the Ioffe Pritchard described below.
The magnetic part of the upper MOT was left unchanged from the work in pvious theses. The optical part of the upper MOT by contrast was completely re-designed to allow the integration of the optical detection system with the MOT (see Section 3.2). This caused new restrictions on the space available for the MOT optics,
33/66 50/50
50/50 50/50
50/50
Vertical (y)
MOT and No repumper
Horizontal (x) MOT + Repumper Horizontal (z) MOT + Repumper Repumper light IN along 2 axis MOT light IN Repumper light IN along 1 axis A C B Linear pol Linear pol Linear pol From TA (side table):
200 mW
From main table: 5 mW
Figure 2.8: The 6-in 6-out fiber distributor or octopus (3 inputs unused) allows beams from different modules to be easily combined and the polarization to be controlled as well as acting as a spatial filter for our MOT beams.
which were satisfied with the introduction of a fiber distribution system from Cana-dian Instrumentation and Research Limited (CIRL), the "octopus" (see Fig. 2.8). This consisted of 6 fiber inputs and 6 outputs and provided an increase in flexibility and convenience as well as savings in space in comparison with the previous bulk optics based system. The distribution system is based on polished evanescent wave couplers. These couplers bring the cores of two fibers close together with their po-larization axes aligned, removing part of the cladding and optically contacting the polished faces. The two cores then behave as if they are contained within the same cladding and coupling from core to core occurs by evanescent wave. The fibers show low loss and back-reflection and are thermally stable. We found that 67% of the light coupled into the octopus at port A was outcoupled from the 6 fibers and that polarization loss as measured with polarization cubes was less than 1% as limited by the cubes. Stability was found to be one part in a thousand over tests lasting several minutes, going up to two parts in a thousand under physical pressure and five parts in a thousand when heated over 40 C. The system had active temperature stabilization via a thermocontrolled array, model 928T (CIRL), and longer time polarization drifts were found to be negligible in the temperature controlled environment of the lab.
Implementation
Adversely, the heads of the fibers were found to be very sensitive to contamination by airborn particles. Outcoupling is readily reduced by 20% or more due to such
MOT beams 16mm 1/e2 CCD 100 100 xyz xyz
x2
x2
xyz MOT MOT Detection In MOT beams 16mm 1/e2 CCD 100 100 xyz xyzx2
x2
xyz MOT MOT Detection InFigure 2.9: Sketch of integration of MOT and imaging optics in horizontal plane. MOT and imaging beams enter the upper vacuum cell via the same 100 mm focal length lenses.
incidents and since the MOT beams form pairs which need to be carefully balanced to maintain the position of the MOT this was a very disruptive feature. Once such an imbalance was observed the fiber heads needed to be polished with a fiber polishing kit (Thorlabs) to regain performance. Even slight imbalances caused changes to the position of formation of the MOT, and particularly to the position of the optical molasses, inhibiting accurate transfer to the magnetic trap. Alignment was thus very important and so each fiber was situated in a XYZ translation stage (Siskiyou).
By making the housing of the fiber rotatable without coupling to translation, it was possible to also balance the powers of opposing beams. This worked because rotating the fiber head rotates the polarization of the outcoupled beam and in combi-nation with the polarization cube in front of the fiber head (see Fig. 2.9) this rotation thus corresponds to varying the power of the beam after the cube.
The four beams in the plane of the optical table were located on separate modules which could be unclamped from the table and separately inspected and aligned. Each module produced a collimated beam of 25 mm diameter at a height of 37.5 mm and the opposing beams could then be aligned by moving the entire module. Aligning the beams in the vertical direction was a particularly cumbersome process and is best understood by considering Fig. 2.10. Due to the small size of the internal gold mirror it was not possible to use collimated beams as in the four horizontal beams, so the divergence of the vertical beams had to be matched as well as possible.
The performance of the 3D MOT was monitored in a number of ways. Its shape was imaged at an angle of 30◦ with respect to the horizontal MOT beams with a
firewire camera (Imaging Source). Position was monitored on a screen via a pinhole camera from above. The fluorescence power was measured by imaging the MOT onto a DET 110 photodiode (Thorlabs). These diagnostics warned of slower loading times, beam misalignment or lasers out of lock. In combination with the 2D MOT with push beam the upper MOT was found to load in 5 s. Its stability was found to depend critically on alignment, particularly at higher atom numbers where exact alignment can lead to a pulsing behaviour. The power used was 20 mW per beam of cooling light at a detuning of two linewidths (12 MHz) and about 1 mW of repumping light.
Figure 2.10: Vertical MOT beams are reflected on a gold mirror inside the vacuum chamber at an angle of 40 degrees to the horizontal. This mirror is 1 cm in diameter. The beam from below must be slightly divergent to be match the size of the horizontal MOT beams at the location of the sample.
2.6
Magnetic trap
The magnetic trap surrounds the upper vacuum cell with two sets of circular coils (the smaller pinch or axial coils and the larger compensation coils) and four radial racetrack shaped coils. This set-up is shown in Fig. 2.11 and has been described in detail previously [38]. It cleverly provides trapping for both the Ioffe-Pritchard (IP) trap and the upper MOT.
2.6.1
Ioffe-Pritchard quadrupole trap
This geometry is known as a Ioffe-Pritchard quadrupole trap. The quadrupole field is produced by the straight bars of the race-track shaped Ioffe coils creating a field which rises linearly with distance from the central axis with gradient α = 3.53 T/m at full current. The field produced by the curved parts of these coils cancel to good approximation on the symmetry axis. The axial confinement is given by a pair of round pinch coils which produce a largely quadratic field with a curvature of β = 266 T/m2, measured at full current. A second pair of axial (compensation) coils are at greater distances from the center of the trap and compensate the offset magnetic field from the pinch coils leaving a tunable residual trap bottom B0of about
10−4 T depending on the current. This combination produces a cigar shaped trap
with the potential at distance ρ α/β from the symmetry axis given by:
U (ρ, z) = μ α2ρ2+ (B 0+12βz2) 2 , (2.2)
Figure 2.11: Schematic (exploded) of the various coils used in the experiment: Pinch coils (1); compensation coils (2); and Ioffe coils (3) together form the Ioffe Pritchard trap. The additional coils axial PCB coils (4) TOP coils (5) and Axial wire coils (6) were added to the IP trap to provide additional functionality.
where μ =gFmFμBis the magnetic moment of the atom. Taking a harmonic
approx-imation gives the trapping frequencies
ωρ= gFmFμB m ( α2 B0− 1 2β) (2.3) and ωz= gFmFμBβ/m (2.4)
for the radial and axial directions, respectively. It should be noted that because the magnetic moment μ is proportional to the mF number, atoms in the |1, −1 level
are trapped a factor of√2less tightly than in |2, 2 . Typical values of B0 are about
1 G, giving a confinement which is far tighter at full current in the radial direction than in the axial direction. The confinement increases as B0 is reduced, with the
harmonic approximation breaking down at B0 = 0 where the trap becomes linear
in the radial direction. The adjustable current through the compensation coils also gives the option of B0 < 0 which leads to a trap with two minima (see Fig. 2.13)
exploited in [35]. The axial symmetry assumed above is broken by the acceleration due to gravity in the vertical direction. In the harmonic approximation, the main effect of this is a displacement Δy in the vertical direction of the minimum of the potential. The size of this displacement can be simply estimated by equating the potential energy in the trap
U = mω2y2
to that of the shifted potential without gravity
Uef f = mω2(y− Δy)2 (2.6)
giving a shift of
Δy g/ω2. (2.7)
2.6.2
Circuitry
The circuit which allows the versatility to switch between upper MOT and Ioffe-Pritchard trap is shown in Fig. 2.12. The current through the coils is provided by three Hewlett-Packard HP 6681A power supplies whose current and voltage are controlled via analogue outputs from the control system described in Section 2.9. The current through each path A-E is controlled by circuitry based on IGBT switches (IXYS, model IXGN200N60A). These allow for ramping of currents using the gate voltage of the IGBTs or quick switching on a timescale of 50 μs in combination with the diodes D1-D6 and capacitors C1 and C2.
With path A and C open, power supply A gives current to both pinch coils in the same direction, while B and C give double the current to pinch 1 only, but in the opposite direction to A to form an anti-Helmholtz configuration creating the quadrupole field of the MOT (see Section 2.5). The MOT stage was performed with 40 Ain the axial coils.
Using path E instead of path C (i.e. simultaneously opening the MOT IGBT and closing the Ioffe coils IGBT) runs current through all power supplies and coils in series, changing the pinch coils to Helmholtz configuration and giving the Ioffe Pritchard field geometry (see Section 2.6.1) created by the Ioffe coils in the radial directions and pinch coils and compensation coils in the axial direction. This switch could be made in 300 μs. Switching off the Ioffe coils from full current (400 A) before imaging occurred in 60 μs. Path D via the Ioffe bypass IGBT allows one of the Ioffe coils to be bypassed, the lower coil in the vertical direction. This can be used to counteract the influence of gravity which can shift the axis of the axis of the trap in the vertical direction.
During the compression stage the currents of the power supplies were ramped up to the full 400 A of the power supplies. The coils are all water-cooled, as is the control unit consisting of the banks of IGBTs and diodes. Adjusting the gate voltage to the IGBT for the application of ramps from fully open to fully closed allows smooth changes in trap geometry such as compression of the trap. These switches are not designed for longer operation or higher currents at intermediate gate voltages and it was essential that, for instance the single IGBT Ioffe bypass was not used for long and that it’s gate voltage was quickly ramped down as currents were ramped up during compression. Glitches and errors in routines meant that this was not always the case and IGBTs were sometimes destroyed.
Due to a lack of exact replacements of these IGBTs it was necessary to replace some with switches of lower current rating and the magnetic trap loading routines had to be altered accordingly. In the case of the Ioffe bypass IGBT, the original functionality could not be maintained as the replacement IGBT, could not maintain the current sent through path D (see Fig. 2.12) during the transfer from MOT to magnetic trap. This severely impacted the process of transfer, making it a much more delicate process. This bypass ensured that the vertical position of the MOT in the center of the trap coincided with the position of the axis of the IP trap. Without this possibility the MOT had to be released at a position above that of the subsequent axis of the IP trap and then allowed to free-fall the required amount of time so that when the IP trap was turned on, the cloud had fallen to precisely the center of the trap. Failure to time this correctly resulted in a sloshing motion in the trap, heating and consequently atom loss.
A number of positive upgrades were made to the experiment upon re-installation of the magnetic coils in order to mitigate some known problems. Firstly, in its pre-vious incarnation at AMOLF, the magnetic trap when switched off, was observed to make a large acoustic disturbance which propagated via the main table to the stabi-lized lasers and via the bread board to the imaging optics. Secondly, the mounting procedure of the magnetic trap around the quartz cell was lacking control, which meant it was a very precarious task to install or remove the magnetic trap without damaging the quartz cell. The first problem was solved by supporting the coils from the same platform as the vacuum system, whereas all optics were mounted on a sep-arate breadboard. The second problem was solved by installing a guiding structure for vertical adjustments using tightly fitting sliders on support legs.
2.6.3
Additional coils for axial field control
In addition to the main high current coils of the magnetic trap describe above, three sets of coils were added to the trap and used in the experiments here. All are shown in Fig. 2.11. In the axial direction a set of round PCB coils are mounted directly on the compensation coils used to trim the field. Mounted on these PCB coils are an additional set of axial coils consisting of 10 turns of 2 mm diameter copper wire. These were used to adjust the trap bottom B0dynamically in some of our experiments. They
can produce fields up to 40 G but have relatively slow switching times. A set of four racecourse — shaped PCB coils were also mounted on top of the Ioffe bars. These coils have a function crucial to the experiments described in this work, creating a time-averaged orbiting potential (TOP) and are described in the following section.
Another set of extra coils was used to compensate the effect of the earth’s magnetic fields. The entire cell assembly was placed inside a standard six coil earth field compensation cage, providing fields of the order of a few Gauss at the center of the trap with excellent homogeneity over the size of the cloud. This coil system was primarily used to cancel stray magnetic fields. It was also found to be useful to apply
Figure 2.12: Control circuit set-up of magnetic trap. Paths A and C are used during the MOT stage. All coils run in parallel during magnetic trapping with paths A and B deter-mining the axial confinement and offset. The external power supply D with accompanying IGBTs are not used.
homogeneous fields during optical pumping and imaging to provide the desired axis of quantization.
2.7
TOP trap
The time-averaged orbiting potential or TOP trap is a method employed to avoid Ma-jorana losses in traps with a zero trap bottom and was used in the first experimental realization of BEC [14]. A bias field shifts the minimum of the potential away from the center of the trap where the atoms are located. Before the atoms can move to this new trap bottom, the minimum point is moved by rotating the bias field quickly about the center of the original trap at a radius dependent on the size of the bias field. This rotating bias field is known as the TOP field and the radius at which the trap minimum is rotated is known as the radius of death. The atoms remain trapped if the size of the cloud is significantly smaller than this radius and the rotation frequency of the TOP field is much larger than the trap frequencies of the original potential. In the latter case the resulting effective potential can be calculated by taking a time average of the combined potential. The minimum of this effective potential in general does not coincide with the minimum of the instantaneous potential.
In previous work on this apparatus the coils denoted as TOP coils in Fig. 2.11 were used to add time-averaged fields to the Ioffe Pritchard trap in order to create a double trap potential [35]. The TOP coils are connected pairwise in series to produce two orthogonal near-homogeneous fields perpendicular to the trap axis. They consist of just 2 windings so that they have a low inductance and can be switched quickly on a microsecond timescale. The field produced by such a PCB coil pair provides a field with a magnitude of 0.221 GA−1and is supplied with currents up to 5 A.
In the case of the Ioffe-Pritchard trap used in these experiments, the addition of the TOP modulation field Bmto a Ioffe Pritchard trap with B0< 0, shifts the double
zero-points of the potential away from the atoms and ensures that Majorana losses in the double well trap are minimized [35]. The resulting effective potential is shown on the bottom row of Fig. 2.13. Plot (e) of this figure shows the effective double wells in the axial direction while plot (f) shows both the instantaneous potential (dotted line) and effective potential (solid line) in the radial direction.
For B0 > 0, the addition of a TOP field also leads to an effective trap potential
with a higher effective trap bottom ¯B0= (B02+ Bm2)1/2 and a lower radial trapping
frequency than the instantaneous potential corresponding to the Ioffe-Pritchard trap
Ωρ= μα2 m ¯B0 (1−1 2B2m/ ¯B02). (2.8)
These expressions are derived in Chapter 5. It can be seen from Eq. (2.8) that the effective radial trap frequency is dependent on the amplitude of the TOP field Bm. An elliptical effective potential can be created by applying different field strengths
r
r r
z z z
|B| |B| |B|
|B| |B| |B|
Figure 2.13: Trapping geometry of the Ioffe Pritchard trap in the radial (lower) and axial (upper) directions. From left to rights the plots show the situation for B0> 0, B0< 0 and
B0< 0 with the addition of the TOP trap (Bm= 0) .
in x and y directions, breaking the radial symmetry. A time variation in Bm will
produce a non-constant effective potential. By adding a rotating elliptical TOP field, the total effective potential also takes the shape of a rotating ellipse and thus the cloud can be stirred about the z axis.
2.8
RF evaporation
The standard method to cool a quantum gas that is confined in a magnetic trap to the degeneracy temperature is to use radio-frequency (rf)-induced evaporative cooling. The principle of this cooling method is simple: the rf-field causes transitions from trapped to untrapped states by inducing spin flips. These flips occur only at positions in space r where the magnetic field B(r) corresponds to the resonance value of the rf field given
ωrf = μBgF|B(r)|, (2.9)
where ωrfis the frequency of the rf field applied. This allows the preferential removal
of higher energy atoms which are located higher in the potential. Following each removal of atoms, rethermalization via collisions must be possible so that the ensemble of atoms can reach a lower temperature as a consequence. The RF field also effects the magnetic trapping potential so that the effective trap frequencies are reduced. This process is known as the dressed state potential. In our experiments, as the RF frequency is ramped down so is the RF power so that we avoid the problem of significant rf dressing with the result that our trap bottom is within 5 −10 kHz [44] of the value which removes all atoms. The total time of the evaporation became slightly longer because of this, taking a total of up to 15 s to reach the trap minimum (see
Figure 2.14: Typical RF evaporation sequence showing ramp down of DDS frequency (dashed line) and power via voltage controlled attenuator (solid line). Between 29 MHz and 23 MHz, the speed of rampdown is slowed and power attenuated as required by a resonance at this frequency. The power is again ramped down near the final frequency to avoid complications of RF dressing.
Fig. 2.14). The RF (radio-frequency) coil has a diameter of 31 mm and is located between the axial coils of the IP trap (as shown in Fig. 2.11) at a distance 16 mm from the center of the cloud. The current to the coil is controlled by an Agilent DDS from the control rack shown in Fig. 2.15. This produces a frequency with constant voltage which can then be attenuated by an inhouse designed voltage controlled attenuator (VCA) and amplified by Amplifier Research broadband 25 W amplifier with a range from 250 MHz down to 10 kHz, ideal for our purposes. To produce an arbitrary ramp, the DDS is sent a signal which controls the frequency and the VCA a control for the amplitude. A strong resonance was found for RF in the range 4−4.5 MHz about 14 dB higher than surrounding frequencies. This resonance is possibly a mutual inductance effect with the magnet trap [38] and was dealt with by adding extra attenuation at this stage of frequency sweeps (see Fig. 2.14).
2.9
Control
Our experimental control system consists of three levels centered on a PXI 1006 18 slot chassis from National Instruments and was developed entirely at the FOM institute AMOLF. A PCI-MXI-3 card is located in the personal computer and a PXI-MXI-3 card is placed in the chassis. Communication between these cards is made via fiber optic cable. A further 12 slots of the chassis are occupied, with four slots taken by PXI DIO64 (viewpoint systems) cards with Labview interface software providing digital outputs, seven analogue output devices NI 6713 and one analogue input device NI PXI-6070E. Each of these cards is in turn connected to the signal condition rack; a 19 inch rack with various functionality. Two of the digital PXI DIO64 cards are
AI AOTTL TTLAO AO AOSDAO DDS
Signal condition rack
M C S out ch1 ch2 ch3 ch4 M C S out ch 1 or ch 2 ch 3 or ch 4 TTI 1 2 M C S out HP 2 OUT Ext VCO trig in gate sync
HP 1 OUT
Ext VCO trig in gate sync
M C S out
AI AOTTL TTLAO AO AOSDAO DDS
Signal condition rack
Krohn-Hite cos ș sin ș inverter Function Generators Multipliers A cos Ȧt sin ș + B sin Ȧt cos ș A cos Ȧt cos ș - B sin Ȧt sin ș A cos Ȧt B sin Ȧt gate Ȉ XY X=M-offset Y=C-offset Input M (offset) Input C (offset) + + -Input S XY+S +1 Output + + AD835 3
Figure 2.15: Controls for TOP coil experiments: (1) the AD835AN multiplier used in both sets of experiments; (2) multiplication of fast oscillating signals to create a rotating elliptical potential used to stir the cloud in the experiments of Chapter 4; (3) digitally generated oscillations are switched by multiplying with TTL inputs to quickly induce phase changes in Chapter 5.
used to supply a total of 4 banks of digital outputs at TTL level; another is used for a bank of shutter drivers (higher voltage) and the last supplies a pair of DDS systems. The NI PXI-6713 card is a 12-Bit, 1 MS/s (mega-samples per second) per channel analogue output board used for unipolar and bipolar control voltages up to 10 V or in one case to give out preprogrammed waveforms. The NI PXI-6070E is a 12-Bit, 1.25 MS/s16 analogue input multifunction DAQ card and was used for monitoring purposes and occasionally for simple feedbacks to the Labview software.
This control system was responsible for executing the experimental sequence. The cards are run with MXI software for windows on the control PC. The program Measurement and Automation explorer (MAX) provides access and troubleshooting capability to the devices, which can then be controlled with Labview software. There was in addition a card for image acquisition via WinView and image taking was also integrated in the Labview program. Getting all these boards to run in parallel and produce the 250 or so outputs necessary to make each BEC required careful assignation of individual IRQs and DMAs to prevent conflicts caused by sharing. In order to maximize the number of IRQs, all but one USB slot of the personal computer was disabled. In addition it was necessary to run NI-DAQ (traditional) data acquisition software alongside the newer NI-DAQ to deal with the differing generations of components. This prevented the classic communication error "not handshaking" from reoccurring.
A dedicated Labview software state engine also written at the FOM institute AMOLF was capable of running all events of the experiments so that no development of software was required during the period of the experiments. The experimenter had only to devise a timeline which could be set to carry out all the actions necessary for each experimental run. Once the timeline has been executed the experimental data takes the form of a cloud absorption image (see section 3.2). Image manipulation and conversion as well as subsequent data analysis was carried out on a Linux machine using open-source software and routines adapted from those employed on neighboring experiments [49]. The Linux operating system greatly simplified batch processing and had the added advantage of allowing remote data analysis of images.
The control system also facilitated the adaptability needed to control the current through the TOP coils as shown in Fig. 2.15. Signal multiplier boxes were designed inhouse based on the Ad835AN chip, a 250 MHz, Voltage Output 4-Quadrant Mul-tiplier from Analog Devices. A pair of these mulMul-tipliers were used to multiply the fast frequency output of two synchronized Hewlett-Packard 3310 function generators with the slow frequency outputs of a Krohn-Hite model 4024 oscillator and add the outcomes in order to create a rotating elliptical potential (see Chapter 4). The am-plitudes of the signals was controlled via analogue outputs of the signal conditioning rack while the triggering of the function generators was controlled via the TTL out-puts of the rack. In another set of experiments the multipliers were combined with a TTI 4 channel arbitrary wave generator to produce jumps in the phase of the TOP (see Chapter 5). This was done by assigning two channels of the synthesizer to each
coil pair and setting a given phase difference between the channels. The TTL out-puts of the signal condition rack were then using to simultaneously switch channels for both directions, giving a near instantaneous phase jump in both directions.
